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The Spike [or Victoria University College Review 1961]

Does the Mathematician Transcend God?

page 35

Does the Mathematician Transcend God?

Man conforms to the archetype of his own free will he is constantly choosing himself as he is.'

'It is not when you realize that nothing can help you — religion, pride, anything. It's when you realize that you don't need tiny aid.'

—J. P. Sartre

I Traversed the Dark Wood and entered the Tomb of Silence. I unwrapped the sacred bandages and discovered beneath a plaster-cast of a Baroque face; beneath some battered flowers flung from the speeding sports-car; beneath the roll of blank microfilm discovered on a lonely country road; beneath the Poet's identification papers and a treatise by Heurtebise on the Zone Intermediaire — a remarkable document, the contents of which are here revealed for the first time.. .


There are three Possibilities: the Mathematician is freer than God; God is freer than the Mathematician; and they are both equally free. We will prove by a simple argument that only the first of these possibilities is tenable, and interpreting 'freer than' to mean 'transcending' (in the absence of any other definition of transcending) we will answer the leading question by 'yes'.

We will prove that the concept God' no longer extends to the frontiers of our imagination, and that His attributes, where they are defined, are simply the redundant equivalents of Mathematical expressions. Finally, we will ask, if God is to be thus transformed into a Mathematical Symbol, why retain the idea at all — why not just keep the Mathematics?

Before we outline the main argument, it might be wise to define our terms. A 'Mathematician' is an existent who studies and creates Form. God' is an abstract (non-physical) existent having properties analagous to those of the Universe. The latter definition is justified in the following way: we look at the laws of the Universe, e.g. PV equals a constant, and we say 'This bears the mark of a God'. In other words, we infer His existence (if we care to do this at all of course) from a group of finite, particular laws characteristic of the Universe. Therefore God must have some property in common with the Laws. Hence the definition.

The Argument

Let us deal first with the postulate 'God is freer than the Mathematician'. Now the philosophy `Mathematics' is completely free since the Mathematician can comprehend any concept, with the sole proviso that the concept be self-consistent. (We dismiss as irrelevant the fact that this is not always demonstrable.) Furthermore, although Mathematics comprizes only formal systems, it is not for that matter a formal system itself; the logical manifold of Mathematics is an open one, since the Mathematician is Free to conceive of any formal system at all. So, 'Freedom to create Form page 36 within the bounds of consistency '— that is the philosophy of Mathematics. Therefore if God were freer than the Mathematician He would necessarily possess the attribute of inconsistency. This, by general, agreement, is not one of His properties. Even God cannot comprehend a 'square circle' ! We have now eliminated the first postulate, leaving 'God is as free as the Mathematician', and 'God is less free than the Mathematician 'as remaining possibilities.

Now if the former is true, then there is an identical relation between Religion and Mathematics, and Religion is trivially true. For example, if the relation : 'God is infinity' (so often propounded by the Theologians) is in fact an identity, that is, God is one-and-the-same-thing as infinity, then the denotation 'God' is superfluous — just an unnecessary word in our vocabulary. Again we hear 'God is the Form of the Universe'; but so is Space-Time (to be pronounced very quickly with the words run together). Once again — redundancy.

Since we have eliminated two of the possibilities, we conclude that the remaining one alone is correct ('The Mathematician is freer than God '). We are therefore led to the Important conclusion: The Religious system is more restricted than the Mathematical one. God, in other words, is no longer a Generality, He is a Special Case. The interested reader may wish to note the error on page 2 of The Pious Scientist, by J. K. Feibleman, where it says 'There is nothing that can be said about a religion which cannot be said in it' — which, as Bertrand Russell would say, is a sheer mistake!

Unfortunately, the above argument is symmetrical in the terms God and The Mathematician. That is so say, if we interchanged these terms (with a few trivial alterations) we would have to draw the opposite conclusion. For example, 'God equals infinity' could imply that the word Infinity is superfluous. In order to assert that our argument alone is correct we must justify the crucial statement 'the Mathematician is completely free'. A quotation from J. D'Abro's book The Evolution of Scientific Thought, p. 455, should make this clear: 'Whereas the real space in Physics appears to possess a definite metrics and structure, the geometrical space of mathematics is amorphous. It possesses no intrinsic metrics, no special geometry, no size, and no shape, and the geometry which the mathematician credits to this space is purely a matter of choice.'

The whole point is this: what the Mathematician does not say is 'I wonder if N-Dimensional space exists'. What he does say is 'Let N-Dimensional space exists!' He puts himself, as it were, in the role of creator. He has no sooner to deliver the injunction: 'Let imaginary numbers exist!' than i's and z's bearing humble but tenacious smiles appear on his pages. 'May all circles intersect' he cries, and I's and J's in order to their stations leap.

If Therese Desqueyroux, as Sartre tells us, was free to say 'No', then the Mathematician is free to say 'Yes'. This, to use Sartre's phrase, is true Cartesian freedom, 'infinite, nameless, and without destiny'. He must have had wonderful mathematical insight to use such an expressive phrase; one can immediately see the X, Y and Z axes stretching off to infinity, defining the boundless field of possibilities which is the subject-matter of Mathematics.

page 37

It is a great pity that Descartes himself said, 'You ask who obliged God to create these truths, and I say He was as free to make all the lines drawn from the centre to the circumference not equal, as not to create the world.' (To prove the absurdity of this, all we need do is ask Descartes to define the word 'circumference'.)

One objection to this argument is 'But surely since the Mathematician cannot create these N-Dimensional spaces in a physical form, this "freedom" of his is quite imaginary'. But the objection is utterly irrelevant to the discussion at hand, which is a comparison between the mind of the Mathematician and the mind of God. The Thesis of Reality is merely used as a link between Ourselves and our inference of God's existence: there is no need to give it more importance than that.

A final point which illustrates the utter confusion in the minds of most Theologians is that involving God, Unity and Beauty. 'God is associated with Unity', i.e. with all the physical laws, they say. So far so good. But what if the Laws of Science reveal a greater and greater degree of ugliness and chaos, as for example the Gas Laws do? We might go further and say What guarantee is there that the laws of Physics will be unified?' After all, the Principle of Gauge Invariance suggested by the mathematician Hermann Weyl is a unique opportunity to unify the main equations of Science, but is simply not realized in practice. So surely Wevl's mind is superior to that of God's? One is forced to conclude, as Hume did, that this Universe is a botched attempt at world making.

The last loophole of the Theologist is Mysticism. 'What about Faith, Love and other intangible, non-logical concepts?' he asks. 'Are we not justified in developing religion as an apotheosis of these qualities?' The answer is 'no' as shown by the following argument.

The Elimination of Mysticism

The proof is by Reductio Ad Absurdum. Suppose we wish to extend our logical philosophy to include mysticism. Suppose further that all men could be divided into two classes at any instant: the Mystics and the Logicians. To give meaning to this distinction, suppose that to the Mystics only mysticism is comprehensible, and to the Logicians only the world of reason is comprehensible, at any instant. Now the Logicians do not admit any such category of thought as mysticism, and the Mystics cannot form the division, as the act of division is a logical process (which, by hypothesis, they are not capable of).

Therefore no man can know which class he belongs in, which (unless one is prepared to invoke the existence of per se entities) means that no such division exists. That is, the original extension was unjustified.

The only interesting objection to this argument is that it is an argument: that is, in rejecting mysticism by a logical process, viz. an argument, we have simply begged the question. But this is nonsense, because the person proposing the objection is by hypothesis either a mystic or a logician: if he is a mystic his philosophy is inconsistent because he rejects the argument with a reason (begging the question) : if he is a logician, his philosophy is inconsistent, because he would not then have objected page 38 to our using a logical process. So to avoid being charged with inconsistency, the mystic can only maintain a pained silence, which of course suits the logicians down to the ground!

The manuscript becomes rather difficult to follow beyond this stage. There is an obscure reference to 'Man enlarging his ego to include the whole world', and another section, some parts of which are badly faded. . .

'Where do " we " come to an end — at our fingertips ? or at the fingertips of our children ? . . . or perhaps the boundaries of our personalities are purely a matter of definition ... but involving a restorative return to the primitive sources of Being...'

The Infra-Red Laboratory is at present trying to force a meaning out of these latter sections. Madame de S—— believes they refer to the blank walls which have no shadows.

So much for mysticism, so much for metaphysics, so much for God. Camera fades and pans to left, revealing Hemingway-type hero clutching whiskey bottle and Geometry textbook, and gazing in the direction of Mount Egmont.

Michael Heine